• Aechtner, M., N. K.-R. Kevlahan, and T. Dubos, 2015: A conservative adaptive wavelet method for the shallow water equations on the sphere. Quart. J. Roy. Meteor. Soc., 141, 17121726, https://doi.org/10.1002/qj.2473.

    • Search Google Scholar
    • Export Citation
  • Afanasyev, Y. D., and J. D. C. Craig, 2013: Rotating shallow water turbulence: Experiments with altimetry. Phys. Fluids, 25, 106603, https://doi.org/10.1063/1.4826477.

  • Batchelor, G. K., 1969: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids, 12, II-233–II-239, https://doi.org/10.1063/1.1692443.

    • Search Google Scholar
    • Export Citation
  • Beron-Vera, F. J., 2021: Multilayer shallow-water model with stratification and shear. Rev. Mex. Fis., 67, 351364, https://doi.org/10.31349/RevMexFis.67.351.

    • Search Google Scholar
    • Export Citation
  • Bleck, R., S. Benjamin, J. Lee, and A. E. MacDonald, 2010: On the use of an adaptive, hybrid-isentropic vertical coordinate in global atmospheric modeling. Mon. Wea. Rev., 138, 21882210, https://doi.org/10.1175/2009MWR3103.1.

    • Search Google Scholar
    • Export Citation
  • Boffetta, G., and S. Musacchio, 2010: Evidence for the double cascade scenario in two-dimensional turbulence. Phys. Rev. E, 82, 016307, https://doi.org/10.1103/PhysRevE.82.016307.

  • Callies, J., and R. Ferrari, 2013: Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1–200 km). J. Phys. Oceanogr., 43, 24562474, https://doi.org/10.1175/JPO-D-13-063.1.

    • Search Google Scholar
    • Export Citation
  • Cho, J. Y.-K., and L. M. Polvani, 1996: The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere. Phys. Fluids, 8, 15311552, https://doi.org/10.1063/1.868929.

    • Search Google Scholar
    • Export Citation
  • Cox, M. D., and K. Bryan, 1984: A numerical model of the ventilated thermocline. J. Phys. Oceanogr., 14, 674687, https://doi.org/10.1175/1520-0485(1984)014<0674:ANMOTV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Dossa, A. N., A. C. Silva, A. Chaigneau, G. Eldin, M. Araujo, and A. Bertrand, 2021: Near-surface western boundary circulation off northeast Brazil. Prog. Oceanogr., 190, 102475, https://doi.org/10.1016/j.pocean.2020.102475.

  • Dubos, T., and N. K.-R. Kevlahan, 2013: A conservative adaptive wavelet method for the shallow water equations on staggered grids. Quart. J. Roy. Meteor. Soc., 139, 19972020, https://doi.org/10.1002/qj.2097.

    • Search Google Scholar
    • Export Citation
  • Dubos, T., S. Dubey, M. Tort, R. Mittal, Y. Meurdesoif, and F. Hourdin, 2015: DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility. Geosci. Model Dev., 8, 3131–3150, https://doi.org/10.5194/gmd-8-3131-2015.

    • Search Google Scholar
    • Export Citation
  • Favier, B., C. Guervilly, and E. Knobloch, 2019: Subcritical turbulent condensate in rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech., 864, R1, https://doi.org/10.1017/jfm.2019.58.

    • Search Google Scholar
    • Export Citation
  • Ferreira, D., J. Marshall, and J.-M. Campin, 2010: Localization of deep water formation: Role of atmospheric moisture transport and geometrical constraints on ocean circulation. J. Climate, 23, 14561476, https://doi.org/10.1175/2009JCLI3197.1.

    • Search Google Scholar
    • Export Citation
  • Ferreira, D., J. Marshall, and B. Rose, 2011: Climate determinism revisited: Multiple equilibria in a complex climate model. J. Climate, 24, 9921012, https://doi.org/10.1175/2010JCLI3580.1.

    • Search Google Scholar
    • Export Citation
  • Flierl, G.R., and C. S. Davis, 1993: Biological effects of Gulf Stream meandering. J. Mar. Res., 51, 529560, https://doi.org/10.1357/0022240933224016.

    • Search Google Scholar
    • Export Citation
  • Ford, R., 1994: Gravity wave radiation from vortex trains in rotating shallow water. J. Fluid Mech., 281, 81–118, https://doi.org/10.1017/S0022112094003046.

  • Fu, L.-L., and G. R. Flierl, 1980: Nonlinear energy and enstrophy transfers in a realistically stratified ocean. Dyn. Atmos. Oceans, 4, 219246, https://doi.org/10.1016/0377-0265(80)90029-9.

    • Search Google Scholar
    • Export Citation
  • Garrett, C. J., and W. Munk, 1972: Space–time scales of internal waves. Geophys. Astrophys. Fluid Dyn., 3, 225264, https://doi.org/10.1080/03091927208236082.

    • Search Google Scholar
    • Export Citation
  • Garrett, C. J., and W. Munk, 1975: Space–time scales of internal waves: A progress report. J. Geophys. Res., 80, 291297, https://doi.org/10.1029/JC080i003p00291.

    • Search Google Scholar
    • Export Citation
  • Grant, H. L., R. W. Stewart, and A. Moilliet, 1962: Turbulence spectra from a tidal channel. J. Fluid Mech., 12, 241268, https://doi.org/10.1017/S002211206200018X.

    • Search Google Scholar
    • Export Citation
  • Jackson, L., C. W. Hughes, and R. G. Williams, 2006: Topographic control of basin and channel flows: The role of bottom pressure torques and friction. J. Phys. Oceanogr., 36, 17861805, https://doi.org/10.1175/JPO2936.1.

    • Search Google Scholar
    • Export Citation
  • Julien, K., A. Rubio, I. Grooms, and E. Knobloch, 2012: Statistical and physical balances in low Rossby number Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn., 106, 392428, https://doi.org/10.1080/03091929.2012.696109.

    • Search Google Scholar
    • Export Citation
  • Julien, K., E. Knobloch, and M. Plumley, 2018: Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection. J. Fluid Mech., 837, R4, https://doi.org/10.1017/jfm.2017.894.

    • Search Google Scholar
    • Export Citation
  • Kevlahan, N. K.-R., 2021: Adaptive wavelet methods for Earth systems modelling. Fluids, 6, 236, https://doi.org/10.3390/fluids6070236.

  • Kevlahan, N. K.-R., and T. Dubos, 2019: Wavetrisk-1.0: An adaptive wavelet hydrostatic dynamical core. Geosci. Model Dev., 12, 49014921, https://doi.org/10.5194/gmd-12-4901-2019.

    • Search Google Scholar
    • Export Citation
  • Kevlahan, N. K.-R., and F. Lemarié, 2022: Wavetrisk-2.1: An adaptive dynamical core for ocean modelling. Geosci. Model Dev., 15, 6521–6539, https://doi.org/10.5194/gmd-15-6521-2022.

    • Search Google Scholar
    • Export Citation
  • Kevlahan, N. K.-R., T. Dubos, and M. Aechtner, 2015: Adaptive wavelet simulation of global ocean dynamics using a new Brinkman volume penalization. Geosci. Model Dev., 8, 3891–3909, https://doi.org/10.5194/gmd-8-3891-2015.

    • Search Google Scholar
    • Export Citation
  • Kitamura, Y., and K. Ishioka, 2007: Equatorial jets in decaying shallow-water turbulence on a rotating sphere. J. Atmos. Sci., 64, 33403353, https://doi.org/10.1175/JAS4015.1.

    • Search Google Scholar
    • Export Citation
  • Kolmogorov, A. N., 1991: The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Proc. Roy. Soc., A434, https://doi.org/10.1098/rspa.1991.0075.

    • Search Google Scholar
    • Export Citation
  • Lapeyre, G., 2017: Surface quasi-geostrophy. Fluids, 2, 7, https://doi.org/10.3390/fluids2010007.

  • Lapeyre, G., and P. Klein, 2006: Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr., 36, 165176, https://doi.org/10.1175/JPO2840.1.

    • Search Google Scholar
    • Export Citation
  • Le Corre, M., J. Gula, and A.-M. Tréguier, 2020: Barotropic vorticity balance of the north Atlantic subpolar gyre in an eddy-resolving model. Ocean Sci., 16, 451468, https://doi.org/10.5194/os-16-451-2020.

    • Search Google Scholar
    • Export Citation
  • Lvov, Y. L., and E. G. Tabak, 2001: Hamiltonian formalism and the Garrett-Munk spectrum of internal waves in the ocean. Phys. Rev. Lett., 87, 168501, https://doi.org/10.1103/PhysRevLett.87.168501.

    • Search Google Scholar
    • Export Citation
  • Madec, G., and NEMO Team, 2016: NEMO ocean engine, version 3.6. Note du Pôle de modélisation de l’Institut Pierre-Simon Laplace 27, 386 pp., https://www.nemo-ocean.eu/wp-content/uploads/NEMO_book.pdf.

  • Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 57535766, https://doi.org/10.1029/96JC02775.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., D. Ferreira, J.-M. Campin, and D. Enderton, 2007: Mean climate and variability of the atmosphere and ocean on an aquaplanet. J. Atmos. Sci., 64, 42704286, https://doi.org/10.1175/2007JAS2226.1.

    • Search Google Scholar
    • Export Citation
  • Molemaker, M. J., J. C. McWilliams, and I. Yavneh, 2005: Baroclinic instability and loss of balance. J. Phys. Oceanogr., 35, 15051517, https://doi.org/10.1175/JPO2770.1.

    • Search Google Scholar
    • Export Citation
  • Morvan, M., X. Carton, P. L’Hégaret, C. de Marez, S. Corréard, and S. Louazel, 2020: On the dynamics of an idealized bottom density current overflowing in a semi-enclosed basin: Mesoscale and submesoscale eddies generation. Geophys. Astrophys. Fluid Dyn., 114, 607630, https://doi.org/10.1080/03091929.2020.1747058.

    • Search Google Scholar
    • Export Citation
  • Munk, W., 1950: On the wind-driven ocean circulation. J. Atmos. Sci., 7, 8093, https://doi.org/10.1175/1520-0469(1950)007<0080:OTWDOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Nadiga, B. T., 2014: Nonlinear evolution of a baroclinic wave and imbalanced dissipation. J. Fluid Mech., 756, 9651006, https://doi.org/10.1017/jfm.2014.464.

    • Search Google Scholar
    • Export Citation
  • Nastrom, G. D., and K. S. Gage, 1985: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci., 42, 950960, https://doi.org/10.1175/1520-0469(1985)042<0950:ACOAWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Perruche, C., P. Rivière, G. Lapeyre, X. Carton, and P. Pondaven, 2011: Effects of surface quasi-geostrophic turbulence on phytoplankton competition and coexistence. J. Mar. Res., 69, 105135, https://doi.org/10.1357/002224011798147606.

    • Search Google Scholar
    • Export Citation
  • Philander, S., 2001: Atlantic Ocean equatorial currents. Encyclopedia of Ocean Sciences, J. H. Steele, K. K. Turekian, and S. A. Thorpe, Eds., Academic Press, 188–191, https://doi.org/10.1006/rwos.2001.0361.

  • Phillips, N. A., 1954: Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus, 6, 273286, https://doi.org/10.1111/j.2153-3490.1954.tb01123.x.

    • Search Google Scholar
    • Export Citation
  • Phillips, N. A., 1963: Geostrophic motion. Rev. Geophys., 1, 123176, https://doi.org/10.1029/RG001i002p00123.

  • Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. Fluid Mech., 69, 417443, https://doi.org/10.1017/S0022112075001504.

  • Rivière, P., A. M. Treguier, and P. Klein, 2004: Effects of bottom friction on nonlinear equilibration of an oceanic baroclinic jet. J. Phys. Oceanogr., 34, 416432, https://doi.org/10.1175/1520-0485(2004)034<0416:EOBFON>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rubio, A. M., K. Julien, E. Knobloch, and J. B. Weiss, 2014: Upscale energy transfer in three-dimensional rapidly rotating turbulent convection. Phys. Rev. Lett., 112, 144501, https://doi.org/10.1103/PhysRevLett.112.144501.

  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Scott, R. K., and L. M. Polvani, 2007: Forced-dissipative shallow-water turbulence on the sphere and the atmospheric circulation of the giant planets. J. Atmos. Sci., 64, 31583176, https://doi.org/10.1175/JAS4003.1.

    • Search Google Scholar
    • Export Citation
  • Smith, K., and G. Vallis, 2001: The scales and equilibration of midocean eddies: Freely evolving flow. J. Phys. Oceanogr., 31, 554571, https://doi.org/10.1175/1520-0485(2001)031<0554:TSAEOM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Steele, J. H., K. K. Turekian, and S. A. Thorpe, Eds., 2001: Encyclopedia of Ocean Sciences. Academic Press, 3399 pp.

  • Stewart, A. L., J. C. McWilliams, and A. Solodoch, 2021: On the role of bottom pressure torques in wind-driven gyres. J. Phys. Oceanogr., 51, 14411464, https://doi.org/10.1175/JPO-D-20-0147.1.

    • Search Google Scholar
    • Export Citation
  • Stommel, H., 1948: The westward intensification of wind-driven ocean currents. Eos, Trans. Amer. Geophys. Union, 29, 202206, https://doi.org/10.1029/TR029i002p00202.

    • Search Google Scholar
    • Export Citation
  • Stommel, H., 2020: The Gulf Stream. University of California Press, 264 pp.

  • Straub, D. N., and B. T. Nadiga, 2014: Energy fluxes in the quasigeostrophic double gyre problem. J. Phys. Oceanogr., 44, 15051522, https://doi.org/10.1175/JPO-D-13-0216.1.

    • Search Google Scholar
    • Export Citation
  • Sukhatme, J., and R. T. Pierrehumbert, 2002: Surface quasigeostrophic turbulence: The study of an active scalar. Chaos, 12, 439450, https://doi.org/10.1063/1.1480758.

    • Search Google Scholar
    • Export Citation
  • Talley, L., 2011: Descriptive Physical Oceanography: An Introduction. Academic Press, 560 pp.

  • Theiss, J., 2004: Equatorward energy cascade, critical latitude, and the predominance of cyclonic vortices in geostrophic turbulence. J. Phys. Oceanogr., 34, 16631678, https://doi.org/10.1175/1520-0485(2004)034<1663:EECCLA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tulloch, R., and K. S. Smith, 2009: Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum. J. Atmos. Sci., 66, 450467, https://doi.org/10.1175/2008JAS2653.1.

    • Search Google Scholar
    • Export Citation
  • Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 1000 pp.

  • Wieczorek, M., and M. Meschede, 2018: SHTools—Tools for working with spherical harmonics. Geochem. Geophys. Geosyst., 19, 25742592, https://doi.org/10.1029/2018GC007529.

    • Search Google Scholar
    • Export Citation
  • Wunsch, C., 1998: The work done by the wind on the oceanic general circulation. J. Phys. Oceanogr., 28, 23322340, https://doi.org/10.1175/1520-0485(1998)028<2332:TWDBTW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Xu, Y., and L.-L. Fu, 2011: Global variability of the wavenumber spectrum of oceanic mesoscale turbulence. J. Phys. Oceanogr., 41, 802809, https://doi.org/10.1175/2010JPO4558.1.

    • Search Google Scholar
    • Export Citation
  • Xu, Y., and L.-L. Fu, 2012: The effects of altimeter instrument noise on the estimation of the wavenumber spectrum of sea surface height. J. Phys. Oceanogr., 42, 22292233, https://doi.org/10.1175/JPO-D-12-0106.1.

    • Search Google Scholar
    • Export Citation
  • Yoden, S., and M. Yamada, 1993: A numerical experiment on two-dimensional decaying turbulence on a rotating sphere. J. Atmos. Sci., 50, 631644, https://doi.org/10.1175/1520-0469(1993)050<0631:ANEOTD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Yuan, L., and K. Hamilton, 1994: Equilibrium dynamics in a forced-dissipative f-plane shallow-water system. J. Fluid Mech., 280, 369394, https://doi.org/10.1017/S0022112094002971.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 486 486 26
Full Text Views 110 110 5
PDF Downloads 157 157 8

Energy Spectra and Vorticity Dynamics in a Two-Layer Shallow Water Ocean Model

Nicholas K.-R. KevlahanaDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada

Search for other papers by Nicholas K.-R. Kevlahan in
Current site
Google Scholar
PubMed
Close
https://orcid.org/0000-0001-7929-7668
and
Francis J. PoulinbDepartment of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

Search for other papers by Francis J. Poulin in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The dynamically adaptive WAVETRISK-OCEAN global model is used to solve one- and two-layer shallow water ocean models of wind-driven western boundary current (WBC) turbulence. When the submesoscale is resolved, both the one-layer simulation and the barotropic mode of the two-layer simulations have an energy spectrum with a power law of −3, while the baroclinic mode has a power law of −5/3 to −2 for a Munk boundary layer. This is consistent with the theoretical prediction for the power laws of the barotropic and baroclinic (buoyancy variance) cascades in surface quasigeostrophic turbulence. The baroclinic mode has about 20% of the energy of the barotropic mode in this case. When a Munk–Stommel boundary layer dominates, both the baroclinic and barotropic modes have a power law of −3. Local energy spectrum analysis reveals that the midlatitude and equatorial jets have different energy spectra and contribute differently to the global energy spectrum. We have therefore shown that adding a single baroclinic mode qualitatively changes WBC turbulence, introducing an energy spectrum component typical of what occurs in stratified three-dimensional ocean flows. This suggests that the first baroclinic mode may be primarily responsible for the submesoscale turbulence energy spectrum of the oceans. Adding more vertical layers, and therefore more baroclinic modes, could strengthen the first baroclinic mode, producing a dual cascade spectrum (−5/3, −3) or (−3, −5/3) similar to that predicted by quasigeostrophic and surface quasigeostrophic models, respectively.

Significance Statement

This research investigates how wind energy is transferred from the largest ocean scales (thousands of kilometers) to the small turbulence scales (a few kilometers or less). We do this by using an idealized model that includes the simplest representation of density stratification. Our main finding is that this simple model captures an essential feature of the energy transfer process. Future work will compare our results to those obtained using ocean models with more realistic stratifications.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Nicholas K.-R. Kevlahan, kevlahan@mcmaster.ca

Abstract

The dynamically adaptive WAVETRISK-OCEAN global model is used to solve one- and two-layer shallow water ocean models of wind-driven western boundary current (WBC) turbulence. When the submesoscale is resolved, both the one-layer simulation and the barotropic mode of the two-layer simulations have an energy spectrum with a power law of −3, while the baroclinic mode has a power law of −5/3 to −2 for a Munk boundary layer. This is consistent with the theoretical prediction for the power laws of the barotropic and baroclinic (buoyancy variance) cascades in surface quasigeostrophic turbulence. The baroclinic mode has about 20% of the energy of the barotropic mode in this case. When a Munk–Stommel boundary layer dominates, both the baroclinic and barotropic modes have a power law of −3. Local energy spectrum analysis reveals that the midlatitude and equatorial jets have different energy spectra and contribute differently to the global energy spectrum. We have therefore shown that adding a single baroclinic mode qualitatively changes WBC turbulence, introducing an energy spectrum component typical of what occurs in stratified three-dimensional ocean flows. This suggests that the first baroclinic mode may be primarily responsible for the submesoscale turbulence energy spectrum of the oceans. Adding more vertical layers, and therefore more baroclinic modes, could strengthen the first baroclinic mode, producing a dual cascade spectrum (−5/3, −3) or (−3, −5/3) similar to that predicted by quasigeostrophic and surface quasigeostrophic models, respectively.

Significance Statement

This research investigates how wind energy is transferred from the largest ocean scales (thousands of kilometers) to the small turbulence scales (a few kilometers or less). We do this by using an idealized model that includes the simplest representation of density stratification. Our main finding is that this simple model captures an essential feature of the energy transfer process. Future work will compare our results to those obtained using ocean models with more realistic stratifications.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Nicholas K.-R. Kevlahan, kevlahan@mcmaster.ca
Save